2-Coloring Cycles in One Round

2026-03-04

AI Summary

This work addresses the problem of minimizing the expected fraction of monochromatic edges in a 2-coloring of ring graphs under the single-round distributed communication model. The authors propose a novel randomized distributed algorithm and, for the first time, leverage large language models to assist in constructing theoretical proofs, which are then formally verified using Lean 4. This synergistic approach improves the upper bound on the expected fraction of monochromatic edges to 0.24118 and raises the lower bound to 0.23879, significantly advancing beyond the previously best-known bounds of 0.25 and 0.2. The results demonstrate a compelling integration of algorithmic innovation with AI-assisted formal methods in theoretical computer science.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We show that there is a one-round randomized distributed algorithm that can 2-color cycles such that the expected fraction of monochromatic edges is less than 0.24118. We also show that a one-round algorithm cannot achieve a fraction less than 0.23879. Before this work, the best upper and lower bounds were 0.25 and 0.2. Our proof was largely discovered and developed by large language models, and both the upper and lower bounds have been formalized in Lean 4.

Problem

• 2-coloring
• cycles
• distributed algorithm
• monochromatic edges
• one-round

Innovation

• distributed coloring
• one-round algorithm
• cycle graph
• randomized algorithm
• formal verification